The Chern-Kaehler-Einstein equations

Question

Let $(M,\omega)$ be a Kaehler manifold with Kaehler form $\omega$ and Ricci form $\rho$. The Chern-Kaehler-Einstein equations are:

$$\omega - \rho = tr(R(\nabla^{Chern}))$$

where $R(\nabla^{Chern})$ is the curvature of the Chern connection $\nabla^{Chern}$ over $TM_{\bf C}$. Have we solutions for black holes of the Chern-Kaehler-Einstein equations?